1. Cut 1 inch off the bottoms of the asparagus and peel the stems. Bring an inch of salted water to a boil in a wide skillet. Add the asparagus and cook them over medium heat until just cooked through, 4 to 5 minutes. Drain and rinse under cold water to stop the cooking process. Put the asparagus on a cloth towel to dry and cool to room temperature.
2. Remove the skin and fat from the chicken. Cutting on the diagonal, thinly slice the meat (the slices will come out unevenly shaped, but it doesn't matter). Slice the tomatoes into thin rounds.
3. To assemble the salad, set some of the asparagus on one side of each plate, and arrange a bed of sliced tomatoes next to the asparagus. Drizzle some oil over the asparagus and the tomatoes, and season them to taste with salt and pepper.
4. Using a vegetable peeler, shave thin slices of Parmesan over the tomatoes. Top them with slices of the chicken. Shave more Parmesan over the chicken, and drizzle more oil on top. Sprinkle with a generous grinding of pepper.